25edo: Difference between revisions
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== Theory == | == Theory == | ||
25edo is a good way to tune the [[blackwood]] temperament, which closes each circle of fifths at five fifths, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4]]) and 7 ([[7/4]]). It also tunes [[sixix]] temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65. | |||
25edo has fifths 18 cents sharp, but its major thirds of 5/4 are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not [[consistent]]. It therefore makes sense to use it as a 2.5.7 [[subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8/7]]'s with the octave, and so tempers out (8/7)<sup>5</sup> / 2 = 16807/16384. It also equates a [[128/125]] [[diesis]] and two [[septimal]] [[tritone]]s of [[7/5]] with the octave, and hence tempers out [[3136/3125]]. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50edo]]. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[trismegistus]] temperament (or [[mavila]] if it is interpreted as [[3/2]]). In fact, it is a convergent to a melodically optimal "golden" tuning of trismegistus or mavila, at around 672.7 cents. | |||
25edo has fifths 18 cents sharp, but its major thirds of 5/4 are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not [[consistent]]. It therefore makes sense to use it as a 2.5.7 [[subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8/7]]'s with the octave, and so tempers out (8/7)<sup>5</sup> / 2 = 16807/16384. It also equates a [[128/125]] [[diesis]] and two [[septimal]] [[tritone]]s of [[7/5]] with the octave, and hence tempers out [[3136/3125]]. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50edo]]. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[mavila]] | |||
If 5/4 and 7/4 are not good enough, it also does 17/16 and 19/16, just like 12edo. In fact, on the [[k*N subgroups|2*25 subgroup]] 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for a wide range of harmony. | If 5/4 and 7/4 are not good enough, it also does 17/16 and 19/16, just like 12edo. In fact, on the [[k*N subgroups|2*25 subgroup]] 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for a wide range of harmony. | ||
Its step of 48{{c}}, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having a very high [[harmonic entropy]]. This is because the harmonic entropy model is usually tuned to reflect the general perception of quarter-tones being the most dissonant intervals. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant. | |||
=== Possible usage in Indonesian music === | === Possible usage in Indonesian music === | ||
Since 25edo contains [[5edo]] as a subset, and it features an [[antidiatonic]] scale generated by the 672 cent fifth, it can theoretically be used to represent Indonesian music in both [[Slendro]] (~5edo) and [[Pelog]] (~antidiatonic scale) tunings. | Since 25edo contains [[5edo]] as a subset, and it features an [[antidiatonic]] scale generated by the 672 cent fifth, it can theoretically be used to represent Indonesian music in both [[Slendro]] (~5edo) and [[Pelog]] (~antidiatonic scale) tunings. However, many tunings of pelog are also better represented by the tuning's native 3-2-6-3-2-3-6 [[omnidiatonic]] scale. | ||
=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|25}} | {{Harmonics in equal|25}} | ||
=== Subsets and supersets === | |||
Since 25 is 5 x 5, 25edo is the smallest composite EDO that doesn't have any intervals in common with [[12edo]]. Doubling 25edo to get [[50edo]] produces a good [[meantone]] tuning. | |||
== Intervals == | == Intervals == | ||
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*based on treating 25-EDO as a 2.9.5.7.33.39.17.19 subgroup; other approaches are possible. | *based on treating 25-EDO as a 2.9.5.7.33.39.17.19 subgroup; other approaches are possible. | ||
25-edo chords can be named with ups and downs, see [[Ups and | 25-edo chords can be named with ups and downs, see [[Ups and downs notation#Chords and Chord Progressions|Ups and downs notation - Chords and Chord Progressions]]. | ||
[[File:25ed2-001.svg|alt=alt : Your browser has no SVG support.]] | [[File:25ed2-001.svg|alt=alt : Your browser has no SVG support.]] | ||
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default [[File:25-EDO_Sagittal.svg]] | default [[File:25-EDO_Sagittal.svg]] | ||
</imagemap> | </imagemap> | ||
=== Second-best fifth (mavila) notation === | |||
{{Mavila}} | |||
== Regular temperament properties == | == Regular temperament properties == | ||
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A subset of the Passion[13] scale. Approximated from the original Akebono I scale of [[12edo]]. | A subset of the Passion[13] scale. Approximated from the original Akebono I scale of [[12edo]]. | ||
; Unfair [[blackwood]][10] | ; Unfair [[blackwood]][10] | ||